Properties | Discussion | Intervals | Z function

Properties

17edt divides 3, the tritave, into 17 equal parts of 111.880 cents each, corresponding to 10.726 edo. It tempers out 245/243 and 16807/15625 in the 7-limit, 77/75 and 1331/1323 in the 11-limit, and 175/169 and 121/117 in the 13-limit. It supports the no-twos temperament tempering out 245/243 and 77/75, which in terms of tritave patent vals could be written 17&21.

17edt is the sixth zeta peak tritave division.

Discussion

17edt is closely related to 13edt, the Bohlen-Pierce division, because they share the feature of tempering out 245/243. Both 13edt and 17edt have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in 13edt is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly in return for gaining a good approximation of 11/9 (given the context of the weak 5/3 and 7/3), which is in fact the size of the large step. However, by the coincidence of the 11-limit commas 17edt tempers out, 5/3 and 11/9 are off by practically the same amount in opposite directions (+10.7 cents and -11.8 cents), leading to an excellent approximation of 55/27 (only 1.1 cents flat), as are 11/9 and 9/7 (-11.8 cents and +12.4 cents),
leading to an excellent approximation of 11/7 (only .6 cents flat) and these sum to 605/189-1.7 cents, which is also a 16/5 which is only .3 cents flat (in addition to equaling 256).

Intervals

degree of 17edt	note name	cents value	cents value octave reduced
0	C	0
1	Db = B#	111.9
2	Eb = C#	223.8
3	D	335.6
4	E	447.5
5	F = D#	559.4
6	Gb = E#	671.3
7	Hb = F#	783.2
8	G	895.1
9	H	1006.9
10	Jb = G#	1118.8
11	Ab = H#	1230.7	30.7
12	J	1342.6	142.6
13	A	1454.5	254.5
14	Bb = J#	1566.3	366.3
15	Cb = A#	1678.2	478.2
16	B	1790.1	590.1
17	C	1902.0	702.0
18		2013.9	813.9
19		2125.8	925.8
20		2237.6	1037.6
21		2349.5	1149.5
22		2461.4	61.4
23		2573.2	173.2
24		2685.2	285.2
25		2797.1	397.1
26		2908.9	508.9
27		3020.8	620.8
28		3132.7	732.7
29		3244.6	844.6
30		3356.5	956.5
31		3468.3	1068.3
32		3580.2	1180.2
33		3692.1	92.1
34		3804.0	204.0

Notes named so that C D E F G H J A B C = Lambda mode

It's a weird coincidence how the schemes of 17edo and 17edt diatonicism are so similar and how their approximations of 9/7 are off by such similar amounts in opposite directions (17edo -11.6 cents and 17edt +12.4 cents).